Power for T-test comparisons of unbalanced cluster exposure studies

Studies of individuals sampled in unbalanced clusters have become common in health services and epidemiological research, but available tools for power/sample size estimation and optimal design are currently limited. This paper presents and illustrates power estimation formulas for t-test comparisons of effect of an exposure at the cluster level on continuous outcomes in unbalanced studies with unequal numbers of clusters and/or unequal numbers of subjects per cluster in each exposure arm. Iterative application of these power formulas obtains minimal sample size needed and/or minimal detectable difference. SAS subroutines to implement these algorithms are given in the Appendices. When feasible, power is optimized by having the same number of clusters in each arm k_A = k_B and (irrespective of numbers of clusters in each arm) the same total number of subjects in each arm n_Ak_A = n_Bk_B. Cost beneficial upper limits for numbers of subjects per cluster may be approximately (5/ρ) - 5 or less where ρ is the intraclass correlation. The methods presented here for simple cluster designs may be extended to some settings involving complex hierarchical weighted cluster samples.